Qualitative and Quantitative Aspects in the Development of Proportional Reasoning

Abstract

An unsolved problem in stage-wise development is the dichotomy between sudden changes when passing from one stage to another, and more gradual changes between stages. This problem is studied here with an experiment on proportional reasoning.

An instrument, the Sharing Cakes Experiment, was devised for group questioning and administered to children between 9 and 16 years of age. It is made up of 24 items, each consisted of the comparison of two ratios presented graphically and included both multiple-choice and open-ended questions.

First-order analysis consisted of scalogram analysis, categorization of items on an ordinal scale, and chronological differentiation of subjects. Five stages were significantly differentiated, with structures integrating one another. Factor analysis performed on the results yielded seven factors which corresponded to the stages found.

A second-order analysis was then applied and consisted in searching for a pattern in the succession of structures: i.e., a structure of structures. Two periods of four phases each were found. These are described in terms of “increasing equilibration,” or “adaptive restructuring” of the problem-solving scheme to new reality. This involves a “dialectical” interchange between subject and environment when the scheme is confronted with a new variable in a problem, with integration of novelty by the scheme.